theorem
  dual S = {Y:Y` in S}
proof
  thus dual S c= {Y:Y` in S}
  proof
    let X1 be object such that
A1: X1 in dual S;
    reconsider Y1=X1 as Subset of X by A1;
    Y1` in S by A1,SETFAM_1:def 7;
    hence thesis;
  end;
  let X1 be object;
  assume X1 in {Y:Y` in S};
  then ex Y st Y=X1 & Y` in S;
  hence thesis by SETFAM_1:def 7;
end;
